15 problems

Let $G$ be the Paley graph of prime order $p$, namely $G=\operatorname{Cay}(\mathbb{Z}_p,S)$ where $S$ is the set of quadratic residues modulo $p$. Let $a,b\in\mathbb{Z}_p^*$ have…

McDonald–Sahay–Wyman's conjecture. As $N\to\infty$ through the primes,

Let $p\equiv 1\pmod{4}$ be prime, let $\Phi_p$ be the Paley equiangular tight frame, and let $\delta_K=\delta_K(\Phi_p)$ denote its RIP constant of order $K$. Paley ETF RIP-constan…

Let $p$ be a prime for which the quartic residues are defined, let $F_p$ be the induced subgraph of the Paley graph $G_p$ on the nonzero fourth powers modulo $p$, and let $A_{F_p}$…

Let $p$ be prime, let $G_p$ be the Paley graph, let $g$ be a multiplicative generator of $\mathbb F_p^\times$, and let $\vartheta$ denote the Lovász theta function. Improvement con…

Necklace character-sum conjecture. For every $k\geq1$,

Let $G_p$ be the Paley graph of prime order $p$, let $a\geq1$, and let $\mathcal I_a(G_p)$ denote the independent sets of size $a$. For $I\in\mathcal I_a(G_p)$, let $A_{G_{p,I}}$ b…

For a prime $p$, let $G_p$ be the Paley graph, let $\overline G_p$ be its complement, and write $\alpha(G_p)=\omega(\overline G_p)$ for its independence number. Here…

Let $q>7$ be a prime power, and let $\mathcal{P}(q^2)$ denote the Paley graph on the finite field of order $q^2$. Let $\tau(G)$ denote the maximum order of an induced forest in a g…

Paley Shannon-capacity conjecture. There exist constants $a_k,b_k>0$ such that, for every prime $p\equiv1\pmod{k}$,

Paley graph conjecture. For each $0<\alpha\leq1$, there exist $\beta=\beta(\alpha)>0$ and $p(\alpha)>0$ such that $\mathcal{P}(\alpha,\beta)$ holds for every prime $p>p(\alpha)$. T…

In the setting of the preceding discussion, let $M$ be the set of integers $m$ satisfying … and let $A_{m,n}$ be the matrix defined in the paper, with $L(n)$ its associated index s…

Let $p$ be a prime with $p\equiv 1\pmod{4}$, and let $q=p^{2s+1}$ for some positive integer $s$. Write $\omega(P_q)$ for the clique number of the Paley graph $P_q$. Improved Paley-…

Let $G_p$ be the Paley graph for a prime $p\equiv 1~(\operatorname{mod}4)$, let $\operatorname{LS}(p)$ denote the linear programming bound obtained in the paper, and let…

For primes $p\equiv1\bmod4$, let $\chi$ be the Legendre symbol modulo $p$. Define $\operatorname{PaleyDiscrepancy}[\alpha,\beta]$ to mean that, for every sufficiently large $p$, ……